Following Kovid's post,
1. I uninstalled Calibre 1.9 64-bit.
2. I installed Calibre 1.9 32-bit.
3. I started Calibre up, then switched to debug mode as advised by Kovid.
4. I adjusted the size and position of the Calibre window.
5. I opened an e-book and adjusted the viewer window size and position, font, and other niceties.
6. I opened about 6 e-books in succession.

Everything worked beautifully! The program window and the viewer window size and position, etc., were all remembered precisely.
7. I closed Calibre then opened the debug file described by Kovid. Here are the contents:
calibre Debug log
calibre 1.9 isfrozen: True is64bit: False
Windows-7-6.1.7601-SP1 Windows ('32bit', 'WindowsPE')
32bit process running on 64bit windows
('Windows', '7', '6.1.7601')
Python 2.7.5
Windows: ('7', '6.1.7601', 'SP1', 'Multiprocessor Free')
Starting up...
Started up in 23.17 seconds with 494 books
Worker Launch took: 1.50900006294
Worker Launch took: 0.260999917984
Worker Launch took: 0.248999834061
Worker Launch took: 0.286999940872
Worker Launch took: 0.253000020981
Worker Launch took: 0.323999881744
8. I reopened Calibre in normal mode. The program windows size and position I had set were remembered!
9. I then opened 6 different e-books in succession.
Again, everything worked beautifully! The program window and the viewer window size and position, font, etc., were all remembered precisely.
I am dumfounded!

I am happy that everything is working normally again, and am embarrassed

that everything seems to be fixed now, when I didn't do anything other than uninstall the 64-bit version of Calibre and reinstall the 32-bit version. Could running in debug mode have fixed the problem?
Whatever the reason for the fix, I am happy and pleased with Calibre once again. It's like getting an old friend back!
My thanks to RB and Kovid for their attention to my plea for help and getting me back to the pleasurable use of this fantastic program!
Frank D